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Wednesday, July 31, 2013

Surviving a Zombie Apocalypse: Just Do the Math

This equation could
spell your doom: (bN)(S/N)Z = bSZ. That is, if you ever found yourself in the
midst of a zombie pandemic.

That's because the
calculation describes the rate of zombie transmission, from one walking dead
individual to many, according to its creators, Robert J. Smith?, a mathematics
professor at the University of Ottawa who spells his name with a "?"
at the end, and his students.

Smith's work has inspired other researchers tocreate zombie mathematical models, which will be published with Smith's work in
the upcoming book, "Mathematical Modeling of Zombies" (University of
Ottawa Press, 2014).

Though of course done
tongue-in-cheek, Smith's study demonstrates why zombies are the viruses of the
monsterworld. Their likeness to viruses makes the creatures ideal subjects for
theoretical epidemiological analyses, which can be used to capture the public's
imagination as well as explore scientific principles, Smith said.

As for a zombie
apocalypse, Smith's model shows that a zombie infection would spread quickly
(with N representing total population, S the number of susceptible people, Z
the zombies, and bthe likelihood of transmission). It also shows that zombies
would overtake the world— there's no chance for a "stable
equilibrium" in which humans could coexist with the undead or eradicate
the disease.

Only coordinated
attacks against the zombies would save humanity, the model shows.

Epidemiology and 'WWZ'

Models of disease
outbreaks, like the one Smith developed, play a prominent role in real-life
epidemiology, Smith said.

"Unlike most
popular monsters, zombies are inherently biological in nature," said Mat
Mogk, founder of the Zombie Research Society. "They don't fly or live
forever, so you can apply real-world biological models to them."

Zombies are walking
representations of a contagion, because they depict flesh-devouring monsters
who spread their affliction by gnawing on the healthy. Some recent zombie
flicks, notably "28 Days Later" and "Zombieland," even
explicitly portray zombieism as a virus.

"A zombie is a bit
like giving a virus legs and teeth," said Ian MacKay, a virologist at the
Australian Infectious Diseases Research Centre, University of Queensland, who
bloggedabout "World War Z." "This is basically a virus taking
over a host, and spreading very quickly and efficiently. … It's an extreme
virus-transmission event, if you like."

In "World War
Z," Brad Pitt plays a U.N. inspectorsearching the globe for the origin of
the zombie outbreak, paralleling the quests of many real-life virus hunters,
Mackay said. "Trying to find the index case, or case zero, bears quite a
resemblance to conventional epidemiology," Mackay said. The movie is
(somewhat loosely) based on Max Brooks' novel of the same name, which included
unprecedented, true-to-life detailabout the political, medical and sociological
ramifications of a zombie outbreak, earning the thriller a spot on a U.S. Naval
War College reading list.

Math tackles the hordes

Analyzing zombies addsa couple of new wrinkles to traditional disease modeling, Smith said: Dead
people can be resurrected as zombies, and humans will attack the infected.
"Usually, the dead aren't a dynamic variable," Smith said. "And
people don't try to kill the people who have an infection."

Those elements —
infections and attacks on zombies — made the model more complicated, because
they introduce two nonlinear factors, or factors that don't change at a
constant rate, said Smith, who has modeled outbreaks of HIV, malaria and West
Nile virus. Most disease models include only one nonlinear element: disease
transmission. Having two nonlinear factors makes zombie math extremely
sensitive to small changes to parameters, Smith said.

The most important
parameter, however, was the infectivity of the zombie disease. In zombie
movies, the affliction spreads fast, Mackay said. In "World War Z,"
for instance, Pitt's character counts out the seconds from bite to
zombification, whereas most infections take days, months or even years in the
case of HIV to manifest.

That high infectivitymakes the zombie epidemic unstoppable in most cases, according to Smith's
model. "Because it only takes one zombie to overtake a city," neither
quarantine nor a slower disease progression could stop the Zombie Apocalypse —
only delay it, Smith said. Only frequent, increasingly effective attacks
against humanity's transformed brethren would win an actual zombie war, he
said.

To model that kind of
human-zombie tangling, Smith used a relatively new mathematical technique
called "impulsive differential equations," which show how abrupt
shocks affect systems. Commonly used to model satellite orbits, the technique
didn't appear until the 1990s, whereas most mathematical tools date back
centuries, Smith said.

Zombies IRL

Applying such
techniques to the flesh-devouring masses provides more than geeky
entertainment, Smith said. It also serves an educational purpose, with a number
of colleges and even high schools using the paper to introduce mathematical
modeling to students, he said. "Teachers say it's the first time they've
gotten their kids interested in math."

Tara Smith, aninfectious disease professor at the University of Iowa, uses the paper to show
how math models can predict the effects of quarantines, vaccines and other
public health measures.

The zombie model's
methods have already proved useful in at least one real-life analysis. While
working on a model of HPV (human papillomavirus), Robert Smith's team noted
that transmission via both gay and straight sex introduced two nonlinear
variables to the equation. Fortunately, the zombie model had already blazed
this path, demonstrating how to handle multiple nonlinear factors.

That real-world
relevance in part explains the pop-culture resurgence of zombies over the last
few years, Mogk said. As epidemics and emerging diseases like SARS and swine
fluhave grabbed the headlines, zombie fictions like "Walking Dead"
and "28 Days Later" have brought the undead a new cultural cachet, he
said.

"With increasing
urbanization, you're getting all these new diseases," he said. "It's
almost a disease of the week or disease of the month now." And those
flesh-hungry viruses-with-teeth are poised to reflect the public's
pandemic-related anxieties.

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